the modulus of the mean velocity vector $$|\left
In accordance with the problem
$$\vec v= v_x \vec t$$, so, $$|< \vec v >|=|< v_x >|$$
Hence, using part $$(a), |< \vec v >|= \left| \dfrac{2 \sqrt 2 a \omega}{3 \pi} \right|=\dfrac{2 \sqrt 2 a \omega}{3 \pi}$$
A string of length 40 cm and weighing 10 g is attached to a spring at one end and to a fixed wall at the other end. The spring has a spring constant of $$160 N m^{-1}$$ and is stretched by 1.0 cm. If a wave pulse is produced on the string near the wall, how much time will it take to reach the spring ?
The equation of a wave travelling on a string stretched along the X-axis is given by $$y = Ae^{\left(\dfrac{x}{a} + \dfrac{t}{T} \right)^2}$$ Find the wave speed.
The equation of a travelling wave is $$y=60\cos (1800\ t-6x)$$ where $$y$$ is in microns, $$t$$ in seconds and $$x$$ in metres. The ratio of maximum particle velocity to velocity of wave propagation is:
the mean value of its velocity vector projection $$\left<\nu_n \right>$$;
A sinusoidal transverse wave of wavelength 20 cm travels along a string in the positive direction of an $$x$$ axis. The displacement $$y$$ of the string particle at $$ x = 0$$ is given in Fig. 16-34 as a function of time $$t$$. The scale of the vertical axis is set by $$y_s = 4.0 \space cm$$. The wave equation is to be in the form $$y\left (x, t \right ) = y_m \sin \left (kx \pm \omega t + \phi \right )$$. (a) At $$t = 0$$, is a plot of $$y$$ versus $$x$$ in the shape of a positive sine function or a negative sine function? What are (b) $$y_m$$ (c) $$k$$ (d) $$\omega$$ (e) $$\phi$$ (f) the sign in front of $$\omega$$ and (g) the speed of the wave? (h) What is the transverse velocity of the particle at $$x = 0$$ when $$t = 5.0 \space s$$?
The linear density of a string is $$1.6 \times 10^{-4} \space kg/m$$. A transverse wave on the string is described by the equation $$y = \left (0.021 \space m \right ) \sin \left [\left (2.0 \space m^{-1} \right ) x + \left (30 \space s^{-1} \right ) t \right ]$$. What are (a) the wave speed and (b) the tension in the string?
A generator at one end of a very long string creates a wave given by $$y=(6.0 \mathrm{cm}) \cos \dfrac{\pi}{2}\left[\left(2.00 \mathrm{m}^{-1}\right) x+\left(8.00 \mathrm{s}^{-1}\right) t\right]$$ and a generator at the other end creates the wave $$y=(6.0 \mathrm{cm}) \cos \dfrac{\pi}{2}\left[\left(2.00 \mathrm{m}^{-1}\right) x-\left(8.00 \mathrm{s}^{-1}\right) t\right]$$ Calculate the speed of each wave.
Use the wave equation to find the speed of a wave given by $$y\left ( x, t \right ) = \left (3.00 \space mm \right ) \sin \left [\left (4.00 \space m^{-1} \right ) x - \left ( 7.00 \space s^{-1} \right )t \right ]$$
Use the wave equation to find the speed of a wave given by $$y\left ( x, t \right ) = \left (2.00 \space mm \right ) \left [\left (20 \space m^{-1} \right ) x - \left (4.00 \space s^{-1} \right )t \right ]^{0.5}$$
A sinusoidal transverse wave traveling in the positive direction of an $$ x $$ axis has an amplitude of $$ 2.0 \mathrm{cm}, $$ a wavelength of $$ 10 \mathrm{cm}, $$ and a frequency of $$ 400 \mathrm{Hz} $$. If the wave equation is of the form $$ y(x, t)=y_{m} \sin (k x \pm \omega t), $$ what is the speed of the wave?